# Part - 2 Lecture - 5 Chapter 1 Relations and Functions

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5. Consider f : N → N, g: N → N and h: N → R defined as f (x) = 2x, g(y) = 3y + 4 and h(z) = sin z, ∀ x, y and z in N. Show that ho(gof) = (hog)of. (N)

6. Find gof and fog, if (N)

(i). \(f(x)=|x| \text{ and }g(x)=|5x-2| \)

(ii). \( f(x)=8x^3 \text{ and } g(x)=x^\frac{1}{3}\)

7. If f : R → R be given by\( f(x)=(3-x^3)^\frac{1}{3}\), then find fof(x). (N)

8. Let\( g,f: R \rightarrow R\) be defined by \(f(x)=2x+1\) and \(g(x)=x^2-2,\forall x\in R \), respectively. Then, find gof. (E)

9. Let \( g,f: R\rightarrow R\) be defined by \(g(x)=\frac{x+2}{3},f(x)=3x-2\) write fog(x). (B)

10. If f : R → R defined by \(f(x)=\frac{x-1}{2}\), find (fof)(x). (B)

11. If f : R → R defined by \(f(x)=x^2-3x+2\), find (fof)(x). (E)

12. If \(f(x)=log{\left(\frac{1+x}{1-x}\right)}\), show that\( f\left(\frac{2x}{1+x^2}\right)=2f(x)\). (B)

13. Let \(f: R \rightarrow R \) be defined by \(f(x)=x^2+1\) , find the pre image of 17 and -3. (B)

14. If \(f: R \rightarrow R, g: R \rightarrow R\), given by \(f(x)=[x],g(x)=|x|\), then find \(fog\left(-\frac{2}{3}\right) \text{ and } gof\left(-\frac{2}{3}\right) \). (B)